Nonlinear Bilevel Programming Problem Research Articles

A nonmonton active interior point trust region algorithm based on CHKS smoothing function for solving nonlinear bilevel programming problems

<abstract><p>In this paper, an approach is suggested to solve nonlinear bilevel programming (NBLP) problems. In the suggested method, we convert the NBLP problem into a standard nonlinear programming problem with complementary constraints by applying the Karush-Kuhn-Tucker condition to the lower-level problem. By using the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, the nonlinear programming problem is successively smoothed. A nonmonton active interior-point trust-region algorithm is introduced to solve the smoothed nonlinear programming problem to obtain an approximately optimal solution to the NBLP problem. Results from simulations on several benchmark problems and a real-world case about a watershed trading decision-making problem show how the effectiveness of the suggested approach in NBLP solution development.</p></abstract>

Hierarchical coordinated energy management strategy for electricity-hydrogen integrated charging stations based on IGDT and hybrid game

The popularity of environment-friendly vehicles promotes the rapid development of electric and hydrogen integrated charging stations (EHI-CSs), which makes the power system face unprecedented challenges. Optimized and effective management strategies are needed to coordinate the operation among EHI-CSs and the distribution system operator (DSO). To this end, a bi-level energy management framework is proposed for DSO and EHI-CSs. The framework takes DSO as the leader of the upper game and provides dynamic differential pricing for EHI-CSs, while EHI-CSs work as the followers, react to the leader's decision in the manner of Nash-Harsanyi bargaining game. Moreover, aiming at the uncertainty of photovoltaic, a risk aversion robustness model for EHI-CSs is designed with information gap decision theory. A two-stage solution is proposed to solve the nonlinear bi-level programming problem with hierarchical energy interaction. Firstly, the lower-level energy cooperation is equivalent to two sub-problems of coalition's profit maximization and payment bargaining, the bisection approach is employed to distributed solve the energy pricing problem in the first stage with tie-line power and prices being the only exchange information. Then the second stage determines the payment according to the scheduling results. Numerical simulations verify the effectiveness of the proposed model and algorithm.

Value chain design through synergistic optimisation of configuration and operation of value activities: an application case of cold chain logistics

The ‘no breakage chain’ characteristics of cold chain logistics make it important to study the synergistic optimisation between its activities. This study analyses the value chain of cold/frozen goods logistics businesses and proposes a value chain-based synergistic framework: intra-enterprise synergy, and synergy of internal and external enterprises. Based on this framework, a cost optimisation model for intra-enterprise synergy of value activities is established; then with consideration of the market-oriented value activity configuration, a synergistic optimisation model for internal and external enterprise between customer utility and enterprise cost through value activity configuration and operations is established. The synergistic optimisation model is a 0–1 nonlinear bilevel programming (BLP) problem, which is solved by using a bilevel nested genetic algorithm (BNGA). Finally, the established synergistic optimisation model is applied to case analysis of a cold chain logistics company as a computational example to demonstrate the effectiveness of the model.

An evolutionary algorithm based on approximation method and related techniques for solving bilevel programming problems.

In the engineering and economic management fields, optimisation models frequently involve different decision-making levels. These are known as multi-level optimisation problems. Because the decision-making process of such problems are hierarchical, they are also called a hierarchical optimisation problems. When the problem involves only two-level decision-making, the corresponding optimisation model is referred to as a bilevel programming problem(BLPP). To address the complex nonlinear bilevel programming problem, in this study, we design an evolutionary algorithm embedded with a surrogate model-that it is a approximation method and correlation coefficients. First, the isodata method is used to group the initial population, and the correlation coefficients of the individuals in each group are determined based on the rank of the leader and follower objective functions. Second, for the offspring individuals produced by the evolutionary operator, the surrogate model is used to approximate the solution of the follower's programming problem, during which the points in the population are screened by combining the correlation coefficients. Finally, a new crossover operator is designed by the spherical search method, which diversifies the generated offspring. The simulation experimental results demonstrate that the proposed algorithm can effectively obtain an optimal solution.

An Active-Set Fischer–Burmeister Trust-Region Algorithm to Solve a Nonlinear Bilevel Optimization Problem

In this paper, the Fischer–Burmeister active-set trust-region (FBACTR) algorithm is introduced to solve the nonlinear bilevel programming problems. In FBACTR algorithm, a Karush–Kuhn–Tucker (KKT) condition is used with the Fischer–Burmeister function to transform a nonlinear bilevel programming (NBLP) problem into an equivalent smooth single objective nonlinear programming problem. To ensure global convergence for the FBACTR algorithm, an active-set strategy is used with a trust-region globalization strategy. The theory of global convergence for the FBACTR algorithm is presented. To clarify the effectiveness of the proposed FBACTR algorithm, applications of mathematical programs with equilibrium constraints are tested.

An interior-point trust-region algorithm to solve a nonlinear bilevel programming problem

<abstract><p>In this paper, a nonlinear bilevel programming (NBLP) problem is transformed into an equivalent smooth single objective nonlinear programming (SONP) problem utilized slack variable with a Karush-Kuhn-Tucker (KKT) condition. To solve the equivalent smooth SONP problem effectively, an interior-point Newton's method with Das scaling matrix is used. This method is locally method and to guarantee convergence from any starting point, a trust-region strategy is used. The proposed algorithm is proved to be stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem.</p> <p>A global convergence theory of the proposed algorithm is introduced and applications to mathematical programs with equilibrium constraints are given to clarify the effectiveness of the proposed approach.</p></abstract>

An active-set with barrier method and trust-region mechanism to solve a nonlinear Bilevel programming problem

<abstract><p>Nonlinear Bilevel programming (NBLP) problem is a hard problem and very difficult to be resolved by using the classical method. In this paper, Karush-Kuhn-Tucker (KKT) condition is used with Fischer-Burmeister function to convert NBLP problem to an equivalent smooth single objective nonlinear programming (SONP) problem. An active-set strategy is used with Barrier method and trust-region technique to solve the smooth SONP problem effectively and guarantee a convergence to optimal solution from any starting point. A global convergence theory for the active-set barrier trust-region (ACBTR) algorithm is studied under five standard assumptions. An applications to mathematical programs are introduced to clarify the effectiveness of ACBTR algorithm. The results show that ACBTR algorithm is stable and capable of generating approximal optimal solution to the NBLP problem.</p></abstract>

An efficient memetic algorithm using approximation scheme for solving nonlinear integer bilevel programming problems

Nonlinear integer bilevel programming problems (NIBLPPs) are mathematical models with hierarchical structure, which are known as strongly NP-hard problems. In general, it is extremely hard to solve this kind of problem because they are always non-convex and non-differentiable, especially when integer constraints are involved. In this manuscript, based on a simplified branch and bound method as well as interpolation technique, a memetic algorithm is developed to solve NIBLPPs. Firstly, the leader's variable values are taken as individuals in populations, for each individual in the initial population, a simplified branch and bound method is adopted to obtain the follower's optimal solutions. Then, in order to reduce the computation cost in frequently solving the follower's problems for lots of offspring generated in evolution, the interpolation method is applied to approximate the solutions to the follower's problem for each individuals in populations. In addition, among these approximated points, only potential better points can be chosen to endure further optimisation procedure, so as to obtain precise optimal solutions to the follower's problems. The simulation results show that the proposed memetic algorithm is efficient in dealing with NIBLPPs.

Multi-Sine Cosine Algorithm for Solving Nonlinear Bilevel Programming Problems

In this paper, multi-sine cosine algorithm (MSCA) is presented to solve nonlinear bilevel programming problems (NBLPPs); where three different populations (completely separate from one another) of sine cosine algorithm (SCA) are used. The first population is used to solve the upper level problem, while the second one is used to solve the lower level problem. In addition, the Kuhn—Tucker conditions are used to transform the bilevel programming problem to constrained optimization problem. This constrained optimization problem is solved by the third population of SCA and if the objective function value equal to zero, the obtained solution from solving the upper and lower levels is feasible. The heuristic algorithm didn’t used only to get the feasible solution because this requires a lot of time and efforts, so we used Kuhn—Tucker conditions to get the feasible solution quickly. Finally, the computational experiments using 14 benchmark problems, taken from the literature demonstrate the effectiveness of the proposed algorithm to solve NBLPPs.

The Estimation of Particle Swarm Distribution Algorithm With Sensitivity Analysis for Solving Nonlinear Bilevel Programming Problems

This paper proposes an estimation of particle swarm distribution algorithm (EPSDA) to solve the nonlinear bilevel programming problem (NBLP) by embedding the estimation distribution algorithm (EDA) into the particle swarm optimization (PSO). One Gaussian function is selected to construct the probability distribution for the superior candidate from the present population before executing the velocity and position update rule at each iteration in PSO. Thus, some new individuals viewed as an offspring population are generated from the probability distribution to replace some inferior particles in the current population for making up a new population. Therefore, this proposed algorithm combines PSO (the local search method) and EDA (the global search method) through updating the population to enhance the efficiency of solving NBLP. In experiments, we select four representative examples for linear, quadratic, nonlinear, high-dimensional nonlinear cases to carry out sensitivity analysis on parameters of the proposed algorithm. The results reveal that linearly decreasing inertia weight and adaptive acceleration coefficients are better than constant parameters. Furthermore, the multivariate Gaussian distribution can achieve better performance compared with the normal distribution. These four examples are also used to compare the performance of EPSDA with ones of separate PSO and EDA by setting constant parameters. The experiments show that EPSDA is better than separate PSO and EDA from both the quality of solution and the computational efficiency. The results on four high-dimensional non-convex nonlinear examples demonstrate feasibility and efficiency of EPSDA to solve the high-dimensional nonlinear bilevel programming from the iteration number and computational time.

An efficient memetic algorithm using approximation scheme for solving nonlinear integer bilevel programming problems

An efficient memetic algorithm using approximation scheme for solving nonlinear integer bilevel programming problems

Evaluation and Sequential Dispatch of Operating Reserve Provided by Air Conditioners Considering Lead–Lag Rebound Effect

Air conditioners (ACs) are widely considered as good candidates to provide operating reserve. Demand response rebound, i.e., the rebound peak of aggregate power, may exist when ACs are controlled by changing the set point temperature. The rebound peak during the recovery period, named lag rebound, may cause significantly higher demand than that prior to the reserve deployment event. The rebound peak during the reserve deployment period, named lead rebound, is rarely considered in previous researches but will constrain the duration time to a short period (e.g., 10 min), which greatly limits the utilization of ACs. This paper proposes an optimal sequential dispatch strategy of ACs to mitigate the lead–lag rebound, and thus realize flexible control of the duration time from minutes to several hours. To quantify the effects of lead–lag rebound, a capacity-time evaluation framework of the operating reserve is developed. On this basis, ACs are grouped to be dispatched in sequence to mitigate the lead–lag rebound. The co-optimization of sequential dispatch on the capacity dimension and time dimension forms a mixed-integer nonlinear bilevel programming problem, in which the consumers’ thermal comfort is also guaranteed. Case studies are conducted to validate the proposed strategy.

Two-echelon Supply Chain Considering Multiple Retailers with Price and Promotional Effort Sensitive Non-linear Demand

This study deals with the effects of a supply chain (SC) with single product, multiple retailers and a manufacturer, where the manufacturer(he) produces lotsize of the product that contains a random portion of imperfect quality item. The imperfect quality products are sold in a secondary shop. The new contribution of this paper is a new non-linear demand function. Demand of the end customers varies with pricing and promotional effort of the rivalry amongst the retailers which can be used for the electronic goods, new lunched products, etc. We investigate the behavior of the supply chain under Manufacturer-Stackelberg(MS), and Retailer-Stackelberg(RS) model structures. The nature of the mentioned models provides great insights to a firm’s manager for achieving optimal strategy in a competitive marketing system. Within the framework of any bilevel decision problem, a leader's decision is influenced by the reaction of his followers. In MS model structure, following the method of replacing the lower level problem with its Kuhn-Tucker optimality condition, we transform the nonlinear bilevel programming problem into a nonlinear programming problem with the complementary slackness constraint condition. The objective of this paper is to determine the optimal selling price and promotional effort of each retailer, while the optimal wholesale price of the perfect quality products are determined by the manufacturer so that the above strategies are maximized. Finally, numerical examples with sensitivity analysis of the key parameters are illustrated to investigate the proposed model.

Integrating link-based discrete credit charging scheme into discrete network design problem

In this paper, we present an optimization model for integrating link-based discrete credit charging scheme into the discrete network design problem, to improve the transport performance from the perspectives of both transport network planning and travel demand management. The proposed model is a mixed-integer nonlinear bilevel programming problem, which includes an upper level problem for the transport authority and a lower level problem for the network users. The lower level sub-model is the traffic network user equilibrium (UE) formulation for a given network design strategy determined by the upper level problem. The network user at the lower level tries to minimize his/her own generalized travel cost (including both the travel time and the value of the credit charged for using the link) by choosing his/her route. While the transport authority at the upper level tries to find the optimal number of lanes and credit charging level with their locations to minimize the total system travel time (or maximize the transportation system performance). A genetic algorithm is used to solve the proposed mixed-integer nonlinear bilevel programming problem. Numerical experiments show the efficiency of the proposed model for traffic congestion mitigation, reveal that interaction effects across the tradable credit scheme and the discrete network design problem which amplify their individual effects. Moreover, the integrated model can achieve better performance than the sequential decision problems.

An Objective Penalty Method for Optimistic Bilevel Programming Problems

In this paper, we consider an optimistic nonlinear bilevel programming problem. Under some conditions, we first show that the sequence of solutions to penalty problems converges to the optimal solution of the original bilevel programming problem. We then present an objective penalty method to solve such a problem. Finally, some numerical experiments are performed to illustrate its feasibility.

A bilevel improved fruit fly optimization algorithm for the nonlinear bilevel programming problem

This paper proposes a bilevel improved fruit fly optimization algorithm (BIFOA) to address the nonlinear bilevel programming problem (NBLPP). Considering the hierarchical nature of the problem, this algorithm is constructed by combining two sole improved fruit fly optimization algorithms. In the proposed algorithm, the lower level problem is treated as a common nonlinear programming problem rather than being transformed into the constraints of the upper level problem. Eventually, 10 test problems are selected involving low-dimensional and high-dimensional problems to evaluate the performance of BIFOA from the aspects of the accuracy and stability of the solutions. The results of extensive numerical experiments and comparisons reveal that the proposed algorithm outperforms the compared algorithms and is significantly better than the methods presented in the literature; the proposed algorithm is an effective and comparable algorithm for NBLPP.

Bi-level programming for supplier selection under quantity discount policy

This paper proposes two models to formulate a Supplier Selection Problem (SSP) in a single-buyer, multi-supplier two-echelon supply chain network. The model coordinates order allocation and supplier selection problems under all-unit quantity discount policy. In this way, bi-level programming is employed to obtain two models: 1) The model with buyer as a leader; 2) The model with vendor as a leader. The resulted nonlinearbi-level programming problems are hard to solve. Therefore, Particle Swarm Optimization (PSO) algorithm is used to deal with the complexity of the model and makes it solvable. Numerical results show that the proposed model is ecient for SSP in compliance with order allocation decision making.

Steady-state optimization of biochemical systems by bi-level programming

A new method is proposed for the steady-state optimization of biochemical systems described by Generalized Mass Action (GMA) models. In this method, a bi-level programming with a two-layer nested structure is established. In this bi-level problem, the upper-level objective is to maximize a flux or a metabolite concentration, and the lower-level objective is to minimize the total sum of metabolite concentrations of biochemical systems. The biological significance of the presented bi-level programming problem is to maximize the production rate or concentration of the desired product under a minimal metabolic cost to the biochemical system. To efficiently solve the above NP-hard, non-convex and nonlinear bi-level programming problem, we reformulate it into a single-level optimization problem by using appropriate transformation strategies. The proposed framework is applied to four case studies and has shown the tractability and effectiveness of the method. A comparison of our proposed method and other methods is also given.

Hybrid Biogeography-based Optimization Algorithm for Solving Nonlinear Bilevel Programming

The article deals with a class of nonlinear bilevel programming problems in which the follower’s objective is a quasi-concave function, and a new hybrid biogeography-based optimization algorithm is proposed to search for the value of the leader’s variables. First, an extreme point searching approach is constructed to obtain the optimal solution of the follower’s programming. An efficient mutation operator is then designed by using Zoutendijk feasible direction method so that it can generate high quality potential offspring. Furthermore, a new fitness function is established that can be easily used to force the individuals moving toward the feasible region and improve the feasible solutions gradually. Under some assumptions, theoretical analysis of the algorithm has been presented. Finally, to verify the effectiveness of the algorithm, numerical experiments on 8 test problems are made and performance analysis verify the proposed algorithm can converge to the global optimal solution of bilevel programming problems.

Optimization of Multi-Period Bilevel Supply Chain Planning for Single Supplier and Single Retailer under Demand Uncertainty

This paper presents the optimization of multi-period bilevel supply chain planning for single supplier and single retailer with leader-follower relationship under demand uncertainty. The multiperiod production planning problem for the supplier and the retailer is formulated as the mixed integer nonlinear bilevel programming problem. In order to improve the total profit, the effects of the replacement of leader-follower relationship are investigated. The effectiveness of the replacement of leader-follower relationship is examined by some numerical experiments. A quantity discount contract with quadratic function is also proposed to maximize the total profit under demand uncertainty.